Mutual Coupling

Coupling

• When two coils ( or more than two coils ) are connected by common magnetic flux, they are called as coupled with each other.

Mutual coupling

• The circuit element used to represent magnetic coupling is known as Mutual coupling. 
• It is donated by symbol M and its unit is Henry. 
• The induced emf in the second coil due to current flows through one coil is related by

        e₂ = M di₁ / dt

• The dot sign indicates direction of current in the coil.

• If the current flows through both coil A and coil B, the induced emf in the two coils A and coil B is

       e1 = L1 ( di₁ / dt ) + M ( di₂ / dt )

       e2 = L2 ( di₂ / dt ) + M ( di₁ / dt )

Inductances are in series : Series aiding

• Figure A shows two coils are connected in series such that the current enter the dot end of the coil A whereas it leaves dot end of the other coil B. 
• This type of coil connection is called as series aiding connection of coil. 
• The total induced emf in coil A and coil B due to self inductance and emf induced due to other coil is given by

e1 = L1 ( di / dt ) + M ( di / dt )

     = [ L1 + M ] ( di / dt ) ……..( 1 )

e2 = L2 ( di / dt ) + M ( di / dt )

     = [ L2 + M ] ( di / dt )…….( 2 )

From equation ( 1 ) and ( 2 )

Now total induced emf e = e1 + e2

            = [ L1 + L2 + M ] ( di / dt )………..( 3 )

 Now e = L ( di / dt )……………….( 4 )

From equation ( 3 ) and ( 4 )

L ( di / dt ) = [ L1 + L2 + M ] ( di / dt )

               L = [ L1 + L2 + 2M ]

Series opposition

• If the coil connection is made such that the current enter the dot end from coil A and enter undotted end of coil B, it is called as series opposition connection. 
• Figure B shows series opposition connection of two coils. The axes of two inductive coils are in the same straight line.

• The coil connection is made such that mutual induced emf opposes the self inducted emf of the coils.

e1 = L1 ( di / dt ) – M ( di / dt )

     = ( L1 – M ) ( di / dt )………. ( 5 )

e2 = L2 ( di / dt ) – M ( di / dt )

     = ( L2 – M ) ( di / dt )………….( 6 )

From equation ( 5 ) and ( 6 )

Now total induced emf e = e1 + e2

           = [ L1 + L2 – M ] ( di / dt )………..( 7 )

Now e = L ( di / dt )……………….( 8 )

From equation ( 7 ) and ( 8 )

L = [ L1 + L2 – M ]

If the coils axis are perpendicular to each other, there is no mutual inductance between them therefore total induced emf

e  = e1 + e2

   = L1 ( di / dt ) + L2 ( di / dt )

   = [ L1 + L2 ] ( di / dt )

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